Another set of open questions is whether the (generically invariant for suitable classes of forcings) theories of \(H_{\aleph _2}\)  under bounded category forcing axioms (or under iterated resurrection axioms) isolated in [1]}, [2]} produce model complete theories for some signature extending \(\in _{\Delta _0}\)  and for some theory extending \(\mathsf {ZFC}+\)large cardinals with some \(\Sigma _2\) -sentence not holding
assuming \((*)\text{-}\mathsf {UB}\)  (for example the assertion that canonical functions are not dominating modulo clubs, or some other \(\Sigma _2\) -sentence whose negation can only be forced using a stationary set preserving forcing which cannot be proper).
